# Log-normal distribution, central limit theorem

If $X_1,X_2,...X_N$ are i.i.d. random variables with finite variance. We can say something about the distribution of $Y$ when $Y=X_1X_2...X_N$ by taking the logarithm of both sides: $\log{Y}=\log{X_1}+\log{X_2}+...+\log{X_N}$, if the $\log{X_i}$ are i.i.d random variables with finite variance then $Y$ converges to be normally distributed. My question is how does one show that $f(y)=\frac{1}{y\sigma \sqrt{2\pi}} exp(-\frac{(\log{y}-\mu)^2}{2\sigma^2})$ and then how does one show that $E[Y]=e^{\mu+\sigma^2/2}$

• Does your first statement be $Y = \prod_{1}^{N}$ instead of $Y = \sum_{1}^{N}$. – Satish Ramanathan Jan 14 '16 at 9:48
• You need the $X_i$ to be positive, and the population mean and variance of their logarithms to be $\mu$ and $N \sigma^2$ (the latter of these is slightly forced, so I probably would not ask the question your way). The density and mean of $Y$ then follow from the properties of the log-normal distribution as results from a transformation of variables, for example in math.stackexchange.com/questions/1168355/… – Henry Jan 14 '16 at 11:12